The way I would set about doing it is by making x the subject of the second equation giving us:

x = 8 - 2y

From here we can substitute this into the first equation to obtain:

(8 - 2y)^2 +2y^2=8 which when expanded will give us:

4y^2 - 32y + 64 +2y^2 =8 which we can then simplify to:

6y^2 -32y +56 = 0 From here the quadratic formula can be used to find the values of y (note that there will likely be two) and then from there those values of y can be put into one of the original equations to find the respective values for x